The Computer Search for the Optimal Settings of a Multi-factorial Experiment Using Response Surfaces D-optimality Design Criterion

The Computer Search for the Optimal Settings of a Multi-factorial Experiment Using Response Surfaces D-optimality Design Criterion

Chapter One

OBJECTIVES OF THE STUDY

The main aim and objectives of this dissertation are

(i) to explore and exploit MATLAB 5.3 to derive D-optimal designs for a multi-factorial experiment under each of three alternative second order response surface models,

(ii) to establish D-efficiency and rotatability-efficiency of the optimal designs under each of the three response surface models, and

(iii) to determine the optimum Nitrogen, Phosphorus, Potassium and Sulphur (NPKS) fertilizer applications for grain production of maize, sorghum and millet in Savannah zone of Nigeria.]

CHAPTER TWO

LITERATURE REVIEW

INTRODUCTION

This chapter considers the history and developments in RSM, where several literatures in the area of RSM were reviewed.

THE EMERGENCE AND DEVELOPMENT OF RESPONSE SURFACE METHODS

The genesis of response surface methodology (RSM) can be traced back to the works of J. Wishart, C.P. Winsor, E.A. Mitscherlich, F. Yates, and others in the early 1930s or even earlier. However it was not until 1951 that RSM was formally developed by G.E.P. Box and K.B. Wilson and other colleagues at Imperial Chemical Industries in England, (Box and Wilson, 1951). Their objective was to explore relationships such as those between the yield of a chemical process and a set of input variables presumed to influence the yield. Since the pioneering work of Box and his co-workers, RSM has been successfully used and applied in many diverse fields such as chemical engineering, industrial development and process improvement, agricultural and biological research, even computer simulation, to mention just a few.

The applications of RSM can be found in Edmondson (1991), Smith et al. (1997), Mountzouris et al. (1999a, 1999b, 2001), Regalado et al. (1994), Rosenthal et al. (2001), Kikafunda et al. 1998, Jauregi et al. (1997), Trinca and Gilmour (1998, 1999, 2000a, 2000b, 2001), Gilmour and Ringrose (1999), Gilmour and Trinca (2003), Gilmour and Mead (2003).

However, there are several procedures that are used in the realization of RSM objectives, especially in the area of criteria for choosing a design. These ranges from the least squares based procedures, the integrated mean square error (IMSE) criterion, the theory of design optimality and the design robustness. In our work we focussed mainly on the theory of design optimality and in particular the D-optimality criterion.

Optimal design theory was developed mainly after World War II. Kiefer (1958, 1959, 1960, 1961, 1962a, 1962b) is attributed to having provided the basic mathematical groundwork for optimal design theory. Presently there are two schools of thought regarding the application of the principles of optimal design theory to the derivation of response surface designs: the “Kiefer school” and the “Box school.” In the latter school, bias suspected of being present in the fitted model plays a significant role. In the Kiefer school, however, bias is regarded as insignificant or it does not exist. The main preoccupation in this school is, therefore, with designs that minimize the variance associated with the fitted model, ŷ(x).

The aim of the optimality theory is selecting an optimum experimental design which, in most cases, needs to be multifaceted. Therefore the problem of selecting a suitable design is thus a formidable one (Box and Draper, 1987). The optimum designs are usually constructed using computer algorithm and are therefore referred to as computer-aided designs. One form of the computer-aided designs is the D-optimal designs. It possesses the most important design criterion in applications (Atkinson and Donev, 1992). These types of computer-aided designs are particularly useful when classical designs do not apply. Unlike standard classical designs such as factorials and fractional factorials, D-optimal design matrices are usually not orthogonal and effect estimates are correlated. The D-optimal designs are always an option regardless of the type of model the experimenter wishes to fit (for example, first order, first order plus some interactions, full quadratic, cubic, etc.) or the objective specified for the experiment (for example, screening designs, response surfaces, etc.). The optimality criterion used in generating D-optimal designs is one of maximizing |X’X|, the determinant of the information matrix X’X.

Several researches related to D-optimum designs have been undertaken. Mitchell (1974a, 1974b) in his two papers gave algorithms for the construction of ‘D-optimal’ experimental designs, Mitchell and Bayne (1978) discussed the D-optimal fractions of three-level factorial designs, Galil and Kiefer (1980) in their paper considered time-and space-saving computer methods, related to Mitchell’s DETMAX, for finding D-optimal designs.

CHAPTER THREE

RSM OPTIMIZATION PROCEDURE (PATH OF STEEPEST ASCENT)

INTRODUCTION

In this chapter we considered one of the RSM optimizations procedures i.e. the Path of Steepest Ascent (PSA). The determination of optimal settings of the experimental factors that produce the maximum (or minimum) value of the response in RSM is achieved using the path of steepest ascent as follows.

RSM Experimental Optimisation Procedure (PSA

Planand run a factorial (or fractional factorial) design near/at a starting (x₁⁰ , x₂⁰ ,….)
Fita linear model (excluding interaction or quadratic terms) to the
Determinepath of steepest ascent (PSA) – a quick way to move to the optimum that is gradient
Runtests on PSA until response no longer
Ifcurvature of surface (like quadratic or cubic surfaces) is large go to step (vi), else go to step (i).
Neighbourhoodof optimum – design, run, and fit (using least squares) a second order
Basedon second order model – pick optimal settings of independent

CHAPTER FOUR

COMPUTER-AIDED DESIGNS METHODOLOGY

INTRODUCTON

In this chapter we present the concept of design regions and the conditions necessitating the use of computer-aided designs. The details concerning D-optimal designs, starting from its least squares, properties and sequential algorithms were all discussed. The models to be employed and the measure of rotatability were stated.

DESIGN REGIONS

The common structure to all experiments is the allocation of treatments, or factor combinations to experimental units. Thus, a desirable design of experiments should provide a distribution of points throughout the region of interest, that is, to provide as much information as possible on the problem. The unit (or region of interest), for example, might be a plot of land receiving a unique treatment combination. As is the tradition in RSM, it is convenient for most applications to scale the quantitative factors (variables) which vary between a minimum and a maximum value (x_i,min≤ x_i ≤ x_i,max; i = 1,2,…,k).

CHAPTER FIVE

MATLAB SOFTWARE

INTRODUCTON

In this chapter we considered the MATLAB software which was used in implementing our algorithm for the determination of D-optimal designs.

MATLAB is an acronym for Matrix Laboratory; a powerful fourth generation programming language. It is a high performance language for technical computing it integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include:

Math and computation
Algorithm development
Modelling, simulation, and prototyping
Data analysis, exploration, and visualization
Scientific and engineering graphics
Applicationdevelopment, including graphical user interface

MATLAB is an interactive system and a programming language whose basic data element is an array that does not require dimensioning. This allows solving of many technical computing problems especially those with matrix and vector formulations in a fraction of time it would take to write a program in a scalar non-interactive language such as C or FORTRAN.

CHAPTER SIX

CONSTRUCTION OF D-OPTIMUM DESIGNS

INTRODUCTON

In this chapter we implemented the programs given in chapter five with the aim of generating the D-optimum designs.

Observe that our experiment has 5 levels of nitrogen (N), 5 levels of phosphorus (P), 3 levels of sulphur (S), and 2 levels of potassium (K) giving 5×5×3×2 = 150 treatment combinations. A maximum of the best 21 treatment combinations is required.

The first levels of all the factors are zero levels which are regarded as controls. Without the controls the treatment combinations would be 4×4×2×1= 32. We shall be generating our designs based on with and without controls.

The construction of a D-optimum design for the experiment would be for runs or support points or treatment combinations based on property (6) of the D-optimal designs, which states that, n the number of support point should be p ≤ n ≤ p(p+1)/2, where p is the number of parameter estimates. The procedure for searching for the designs (as described in chapter five), using our program in MATLAB is as follows.

CHAPTER SEVEN

SUMMARY, CONCLUSION AND RECOMMENDATION

INTRODUCTON

In this chapter we present the summary, conclusion and recommendations based on the results obtained in chapter six.

SUMMARY AND CONCLUSION

We recall that the main objective of our thesis is to generate with the aid of computer algorithms an optimal settings for a four factor experiment involving nitrogen, phosphorus, sulphur and potassium at 5,5,3 and 2 levels respectively if controls are used, and 4,4,2 and 1 respectively, if there is no control. The factors are to be tested on maize, sorghum and millet. It should also be noted that in the initial experiment 21 treatment combinations are required because of limitation of resources. In chapter six, we started first by generating the treatment combinations for both with and without controls using the algorithms given in chapter five. We used the exchange algorithm also given in chapter five to search for D-optimal settings (designs) for maize, sorghum and millet using the interactions, quadratic and pure-quadratic models. Since it isn’t possible plotting the optimal settings for with control, the optimal settings generated for without control were then plotted on a three-dimensional space to depict the distributions of the settings, as shown in fig. 6.3 through 6.8.

The three models interaction, quadratic and pure-quadratic models are compared for the crops. The maize was considered separately while sorghum and millet were considered jointly because they have the same initial and optimal settings.

We favoured the use of quadratic model and evaluated the rotatability of the optimal settings (design) of the model using Khuri’s measure of rotatability. It was found that the design was near rotatable with a value of 83.55%.

RECOMMENDATIONS

Based on our observations in 6.6, we recommend the use of the settings generated

by the quadratic model for the initial experiments instead of the ones used. This is because for these type of levels (more than two), the quadratic explains the relationships among factor levels better than the general linear model used for the analysis of the experiment (which only explains relationship of levels for at most two).

The settings used in the initial experiment, where nitrogen, phosphorus, sulphur, and potassium are represented by X1, X2, X3, and X4 respectively are:

Areas of Future Research

This work explores rowexchange algorithms of MATLAB 5.3 to give a procedure for analysing a multi-factor experiment whose design region was of the non-standard type (or restricted). For future research there is the need to consider factors that have three or more levels. Also, the procedure used in generating the treatment combinations for the restricted design region in MATLAB 5.3 is very tedious; there is also the need to devise a simple procedure for this purpose. However, despite the fact that the D-optimal design is established in literature to be the best in applications, there is still the need to compare the performances of the D-optimal designs presented here with the other alphabetic criteria (like A, E, G, V, etc.). One other important area which needs to be explored is the possibility of using the D-optimal designs generated in blocks. This will definitely assist in situations where the factors are qualitative in nature. It will also be of tremendous importance if the D-optimal design criterion can be used to determine optimal settings for a multiresponse experiment.

REFERENCES

Atkinson, A.C. and Donev, A.N. (1992). Optimum Experimental Designs. New York: Oxford.
Bamanga, M.A. and Asiribo, O.E. (2005). “The Use of Response Surface Methodology in the Determination of Optimum Conditions,” Zuma Journal of Pure and Applied Sciences 7(2), pp.174-181.
Box, G.E.P. and Draper, N.R. (1987). Empirical model building and response surfaces, New York: Wiley.
Box, G.E.P., Hunter, W.G., and Hunter, J.S. (1978). Statistics for experimenters. New York: Wiley.
Box, G.E.P. and Behnken, D.W. (1960). “Some New Three Level Designs for the Study of Quantitative Variables,” Technometrics, 2, pp.455-475.
Box, G.E.P. and Draper, N.R. (1975). “Robust Designs,” Biometrika, 62, pp.347-352.
Box, G.E.P. and Hunter, J.S. (1957). “Multifactor Experimental Designs for Exploring Response Surfaces,” Ann. Math. Statist., 28, pp.195-241.
Box, G.E.P. and Wilson, K.B. (1951). “On the Experimental Attainment of Optimum Conditions (with discussion),” J. Roy. Statist. Soc., B13, pp.1-45.
Chigbu, P.E. (1998). “Efficiency, Efficiency factors and Optimality Criteria for Comparing Incomplete-Block Designs,” Journal of Nig. Stat. Assoc., 12, pp.39-55.

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